Discussion and Analysis of the Factors of an Electron Through Measurements with Electron Spin Resonance Kathryn Potter Electron Spin Resonance (ESR) is a very important technique in understanding physics principles at the atomic level. This technique allows observations to be made for the electron’s spin and magnetic moment and most importantly the electron g-factor. This factor connected the basic principles of ESR to the complex principles behind relativistic quantum mechanics. The connection to relativistic quantum mechanics allows an observer to avoid all the underlying complications with making measurements of an electron’s behavior. Therefore, ESR and the values it observes and measured is crucial to understanding the physics concepts occurring at the atomic level. Understanding the laws of physics requires understanding objects and their behaviors at an atomic level. One of these objects is the electron and its spin behavior and magnetic moment. Measuring these factors is extremely rigorous though without relativistic quantum mechanics. When relativistic quantum mechanics is introduced, a factor of 2 is observed for the magnetic moment. The factor of 2 is called the electron gfactor and its value is calculated through the procedure in this paper. This calculation is made by observing a magnetic moment in a static magnetic field precessing at a natural frequency. Resonance can then occur when an oscillating external field is applied at that frequency. It is from this resonance that the procedure makes certain observations and measurements to calculate the g-factor. From this point forward, many advances were made in regards to the electron and its spin component. One of these advances was made when the affect of unpaired electrons in certain materials was measured. This new type of measurement created the concept of electron spin resonance. The theory behind this measurement method is described in the next section. The theory behind the measurement method of the Electron Spin Resonance (ESR) can only be described after first understanding the apparatus used. Figure 1: ESR Diagram The ESR apparatus includes a coil wrapped around a test sample, an RF Oscillator, and an Ammeter. The test sample is placed in a uniform magnetic field and the RF oscillator induces small magnetic fields in the coil at a right angle to the current uniform magnetic field. These two magnetic fields create conditions for resonance, in which measurements are made from. A single electron within the test sample has a spin, which gives it an intrinsic magnetic dipole moment (µS). This relationship is seen in the equation 𝜇𝑆 = 𝑔𝑆 𝜇𝐵 𝑆 ℏ (1) In this equation 𝑔𝑆 is a constant characteristic of the electron (the g-factor); 𝜇𝐵 is the Bohr magneton = 𝑒ℏ/2𝑚𝑒 = 5.788 x 10-9 eV/G (where e is the charge of an electron, me is the mass of an electron, and ℏ is Planck’s constant = 6.582 x 10-16 eV·sec); and S is the spin of the electron. The magnetic dipole moment of an electron is important because it is this property of the electron that interacts with the uniform magnetic field. A second property of the electron, which is briefly described above, is its spin of ± 1/2. Because of these two values for the spin, an electron can only have energies of 𝐸0 ± 𝑔𝑆 𝜇𝐵 /2, where 𝐸0 is the electron’s energy before it is placed in the uniform magnetic field. The difference between these two possible energies gives a value of 𝑔𝑆 𝜇𝐵 B where B is the magnitude of the uniform magnetic field. It is these energy differences, which creates the condition for resonance. Resonance occurs when the frequency (𝜐) of the RF oscillator results in energy equal to 𝑔𝑆 𝜇𝐵 𝐵. The RF oscillator irradiates photons at an energy of ℏ𝜐. At this energy electrons from a lower energy state are able to jump to a higher energy state by absorbing a photon. When this occurs, the test sample is affected, which in turn changes the coil inductance and the RF oscillator oscillations. From these changes an observable change in current is observed and measured. In summary, resonance occurs when the energy of the irradiated photons equals the electron energy differences, which is seen in the equation ℏ𝜐 = 𝑔𝑆 𝜇𝐵 𝐵 (2). B in this equation is calculated from the following equation: 3 4 2 𝜇0 (5) 𝐵= 𝐼 𝑁𝑟 (3) where 𝜇0 = 1.256 x 10-6 v·s/A·m; N is the number of turns in each coil (320); I is the current through each coil; and r is the radius. Equation (2) can be solved for 𝑔𝑆 and broken up more into its components in order to get the following equation: 𝜐 𝑔𝑆 = ( 𝐼 ) ℏ2𝑟 3 (4) 4 𝜇𝐵 𝜇0 ( )2 𝑁 5 However, actually achieving resonance is difficult and requires accurate measurement. The measurement can only be taken when the RF frequency is set at exactly the right value. This value is achieved by adjusting the value of the magnetic field about a constant value. The PASCO ESR Apparatus shown in Figure 1 accomplishes this by sending a small current to the coils around the test sample. This small current allows the magnetic field to vary sinusoidally around a constant value. A dual trace oscilloscope produces a sinusoidal wave demonstrating this value, as seen in Figure 2. Figure 2: ESR on the Oscilloscope The upper sine wave is a function of the current passing through the coils, which is proportional to the magnetic field. The function below this sine wave demonstrates the voltage passing through the RF oscillator. The dips in this function occur when the magnetic field passes the resonance point. These two functions are the main components in the measurement of the ESR. The ESR measurements are still complicated due to various conditions within an atom. These conditions result from complex electromagnetic factors and energy splittings and shifts, which make the ESR measurements less accurate. However, with the PASCO ESR Apparatus these conditions are simplified. The PASCO ESR Apparatus uses a Diphenyl-Picryl-Hydrazyl (DPPH) test sample, which has a simple nature that makes measuring the ESR easier and more accurate. The simple nature of DPPH is due to the fact that is has an orbital angular momentum of zero and only one unpaired electron. These two factors make measurement easier because now for any magnetic field there is only one resulting resonant frequency. Now that the theory behind the measurement method has been explained, the experimental apparatus, setup, and operation method is described in the next section. There are three components to the experimental equipment of the PASCO ESR Apparatus: an ESR Probe Unit, an ESR Basic System, and a Complete ESR System. The equipment comprising each of these components are seen in Figure 3 below. Figure 3: The ESR Apparatus The ESR Probe Unit includes: a Probe Unit with base, three RF Probes and a DPPH sample in a vial, a Passive Resonant Circuit, and a Current Measuring Lead for the Probe Unit. The ESR Basic System includes: an ESR Probe Unit, a pair of Helmholtz coils with bases, and an ESR Adapter. The Complete ESR System includes: an ESR Probe Unit, a pair of Helmholtz coils with bases, and a Control Unit. In addition to these components a dual trace oscilloscope, a DC Ammeter, connecting wires with banana plug connectors, a Frequency Meter, and a Low Voltage AC/DC Power Supply is also used during experimentation. The specific details and technical data for each component is discussed below. The Probe Unit is the most important component of the ESR Apparatus. Figure 4: The ESR Probe Unit This unit allows the frequency and amplitude to be controlled with the two knobs on the top and side of the box. However, the frequency can only be altered within the range of values allowed by the specific RF probe. The different RF probes provide a different range of frequencies, as displayed in Figure 3. This range of frequencies occurs because the inductance of the probe plays a part in determining the inductance of the oscillator circuit. The Probe Unit is plugged directly into the Control Unit when using the Complete ESR System and plugged into an ESR Adapter when using the Basic ESR System. Next, the Control Unit is discussed. The Control Unit allows the Probe Unit to run and be measured with three main functions. Figure 5: The ESR Control Unit First, it provides the Probe Unit and Helmholtz coils with the needed voltage power to run. Second, it measures the RF oscillations of the Probe Unit and provides a reading on the RF Frequency Display screen seen in Figure 5. Third, it delivers two outputs to the dual trace oscilloscope. One output is the RF Signal Output, which is proportional to the RF oscillator current and observes the resonance pulses. The second output is the B-Field Output, which is proportional to the Helmholtz coils current and measures the magnetic field. In addition to these functions, the Control Unit also has a phase shifter knob to make adjustments for the phase shift that occurs between the Helmholtz coils’ current and the voltage the oscilloscope reads. Next, the Helmholtz coils will be discussed. The Helmholtz coils’ main function is to provide a uniform magnetic field for ESR measurement. To measure the ESR the sample material is placed in between these coils to interact with the magnetic field. Figure 6: The Helmholtz Coils As seen in Figure 6, the coils are placed parallel to one another by a separation equal to the value of their radii. These coils create a magnetic field whose value is calculated from equation (3). This discussion of the experimental components now allows the setup of the components to be explained. For this experimental setup, the ESR Complete System with the 30-75 MHz probe is used. All steps of this setup are seen in Figure 7 below. Figure 7: Setup using the Complete ESR System First, the Helmholtz coils are placed 6 cm apart and connected to the Control Unit. The Ammeter is then connected in series between the coils and the Control Unit. Next, the X output of the Control Unit is connected to channel 1 of the dual trace oscilloscope. The oscilloscope is set to a sensitivity of 1 or 2 V/div, a sweep rate of 2 or 5 ms/div, and a coupling of DC and the Umod knob on the Control Unit is set to zero. Then the U0 knob is set at midscale and the Umod knob is adjusted at a clockwise direction in order to increase the AC power of the current in the Helmholtz coils. If this is done correctly, the oscilloscope trace will show a smooth sine wave as seen in Figure 8. Figure 8: Current in Helmholtz Coils This sine wave occurs because the AC magnetic field covers the constant DC field. Next, the Y output of the Control Unit is connected to channel 2 of the dual trace oscilloscope and the oscilloscope is set to a sensitivity of 0.5 or 1 V/div and a coupling of DC. The Control Unit is then connected to the Probe Unit and the RF Probe is placed in the Probe Unit with the DPPH sample. After this step, the Probe Unit is flipped on and the Amplitude knob is set to a medium value. With the Probe Unit on, the Control Unit should provide a reading of the oscillation frequency on the frequency meter. The Probe Unit’s Frequency Control knob is then turned until the Control Unit indicates a frequency output around 50 MHz. Now the Umod knob is set to the 11 o’clock position at approximately the fourth position and U0 is increased to a medium value. These settings ensure the Helmholtz coil’s current has a value around 1 A. After these steps the oscilloscope trace now looks like Figure 9. Figure 9: Traces of the Oscilloscope Display As displayed in Figure 8, the channel 1 trace demonstrates the Helmholtz coil’s current, which is proportional to the magnetic field. In Figure 9, the channel 2 trace demonstrates the voltage passing through the RF oscillator. The pulses in this function correspond to the points of resonance absorption from the magnetic field. If these pulses are not seen, the Umod and RF frequency are adjusted until they become visible. As explained above, a phase shift occurs between the Helmholtz coils’ current and the voltage, and because of this the traces may not be symmetrical like they are in Figure 9. To compensate for this, the phase shifter knob is adjusted until the traces are symmetrical. Now that the setup has been completed, the actual operation of the experiment can begin, which is described in the next section. To start operation, the DC current is adjusted so that the resonance pulses appear when the AC component of the Helmholtz coils’ current is zero. When the DC current is adjusted the resonance pulses move farther or closer to one another. This is accomplished by adjusting the vertical position of trace 2 until the bottom part of the pulse rests on the zero level of trace 1. When this is accomplished the traces will look like Figure 10 below. Figure 10: Final Oscilloscope Display With these adjustments in place, the resonance value is now the DC value indicated by the ammeter and the frequency value is the value displayed on the Control Unit. From the values stated above, the RF frequency and the DC current is measured. For these frequency and current values the magnetic field is measured with a multimeter and the highest value observed is recorded. This step is repeated for several different values of current, and for each value the frequency and magnetic field is measured. Next, the measured value of the magnetic field is compared to the calculated value by using equation (3). However, the value from (3) needs to be divided by 2, to compensate for the coils’ current being set up in parallel, since the current measured is for the total system not just one coil. Finally, the g-factor for an electron of the DPPH sample is calculated from equation (2). With this last calculation the operation procedure is complete. The results acquired from this experimental operation are discussed in the next section. The operation is repeated for five different current values. The measurements gathered for these five different values are displayed in Figure 11 below. Measurement Current (A) Frequency Magnetic Field Magnetic Field (mT) (MHz) (mT) (Calculated) 1 0.70 36.2 1.20 1.67 2 0.80 38.4 1.34 1.92 3 0.90 39.8 1.50 2.16 4 0.99 46.6 1.73 2.37 5 1.00 41.7 1.73 2.39 Figure 11: Measurement Values Several methods are used in order to find the value of the g-factor for an electron of DPPH. First equation (2) is used with the measured frequency and magnetic field. The results for this method are seen in Figure 12. Measurement Frequency (MHz) Magnetic Field (mT) g-factor 1 38.4 1.34 0.343 2 39.8 1.50 0.325 3 46.6 1.73 0.302 4 36.2 1.20 0.306 5 41.7 1.73 0.274 Figure 12: g-factor Results with Measured Magnetic Field Next equation (2) is used with the measured frequency and calculated magnetic field. The results for this method are seen in Figure 13 below. Measurement Frequency (MHz) Magnetic Field (mT) g-factor (Calculated) 1 38.4 1.67 0.247 2 39.8 1.92 0.227 3 46.6 2.16 0.209 4 36.2 2.37 0.224 5 41.7 2.39 0.198 Figure 12: g-factor Results with Calculated Magnetic Field A more accurate way of calculating the g-factor is completed by graphing Current vs. Frequency as seen in Figure 13. Current vs. Frequency 50 45 Frequency (MHz) 40 35 30 y = 26.544x + 17.235 25 20 15 10 5 0 0 0.2 0.4 0.6 Current (A) 0.8 1 1.2 Figure 13: Current vs. Frequency 𝜐 The value of the slope is substituted into equation (4) for 𝐼 and the resulting g-factor value is calculated to be 0.126. However, the currently accepted value for the g-factor is 2. This discrepancy comes from errors with the experiment. These errors most likely occur when setting up the traces on the oscilloscope and measuring the magnetic field. If either step is not completed with pristine accuracy the precision in the g-factor calculation is affected. Because the three methods for calculating the g-factor for this experiment gave values only 0.1 apart, the evidence for the discrepancy stands for measurement error. In conclusion, the g-factor, although difficult to measure, is a very important constant in understanding the behaviors occurring within an atom. One of these behaviors, the resonance resulting from a magnetic field, leads to the measurement of Electron Spin Resonance. This ESR technique allows for many factors within this process to be observed and measured. The observations seen through the oscilloscope provide good understanding as different factors are controlled in order to find the value of the g-factor. This entire procedure and measurement method confirms the usefulness and need of the Electron Spin Resonance.